A d-dimensional framework is a pair (G, p), where \(G=(V,E)\) is a graph and p is a map from V to \({\mathbb {R}}^d\). The length of an edge \(xy\in E\) in (G, p) is the distance between p(x) and p(y). A vertex pair \(\{u,v\}\) of G is said to be globally linked in (G, p) if the distance between p(u) and p(v) is equal to the distance between q(u) and q(v) for every d-dimensional framework (G, q) in which the corresponding edge lengths are the same as in (G, p). We call (G, p) globally rigid in \({\mathbb {R}}^d\) when each vertex pair of G is globally linked in (G, p). A pair \(\{u,v\}\) of vertices of G is said to be weakly globally linked in G in \({\mathbb {R}}^d\) if there exists a generic framework (G, p) in which \(\{u,v\}\) is globally linked. In this paper we first give a sufficient condition for the weak global linkedness of a vertex pair of a \((d+1)\)-connected graph G in \({\mathbb {R}}^d\) and then show that for \(d=2\) it is also necessary. We use this result to obtain a complete characterization of weakly globally linked pairs in graphs in \({\mathbb {R}}^2\), which gives rise to an algorithm for testing weak global linkedness in the plane in \(O(|V|^2)\) time. Our methods lead to a new short proof for the characterization of globally rigid graphs in \({\mathbb {R}}^2\), and further results on weakly globally linked pairs and globally rigid graphs in the plane and in higher dimensions.

d维框架是一对 ( G ,  p )，其中\(G=(V,E)\)是一个图，p是从V到\({\mathbb {R}}^d\)的映射。 ( G ,  p )中边\(xy\in E\)的长度是p ( x ) 和p ( y )之间的距离。如果p ( u ) 和p ( v )之间的距离等于q ( u ) 之间的距离，则G的顶点对\(\{u,v\}\)被称为在 ( G ,  p ) 中全局链接) 和q ( v ) 对于每个d维框架 ( G ,  q ) ，其中相应的边长度与 ( G ,  p )中的相同。当G的每个顶点对在 ( G , p ) 中全局链接时，我们称 ( G ,  p )在\ (  { \mathbb {R}}^d\)中全局刚性。如果存在通用框架（G , p​​​ ​ ) 其中\(\{u,v\}\)是全局链接的。在本文中，我们首先给出\ ({\mathbb {R}}^d\)中\((d+1)\)连通图G的顶点对的弱全局关联性的充分条件，然后证明对于\(d=2\)，这也是必要的。我们使用这个结果来获得\({\mathbb {R}}^2\)图中弱全局链接对的完整表征，这产生了一种用于测试\(O( |V|^2)\)时间。我们的方法为\({\mathbb {R}}^2\)中的全局刚性图的表征提供了一个新的简短证明，并进一步得出了平面和更高维度中的弱全局链接对和全局刚性图。